| L(s) = 1 | + 3-s + 9-s − 2·11-s − 3·13-s + 8·17-s − 19-s + 8·23-s + 27-s + 4·29-s + 3·31-s − 2·33-s + 37-s − 3·39-s − 6·41-s + 11·43-s − 6·47-s + 8·51-s + 12·53-s − 57-s + 4·59-s + 6·61-s + 13·67-s + 8·69-s + 10·71-s − 11·73-s + 3·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.832·13-s + 1.94·17-s − 0.229·19-s + 1.66·23-s + 0.192·27-s + 0.742·29-s + 0.538·31-s − 0.348·33-s + 0.164·37-s − 0.480·39-s − 0.937·41-s + 1.67·43-s − 0.875·47-s + 1.12·51-s + 1.64·53-s − 0.132·57-s + 0.520·59-s + 0.768·61-s + 1.58·67-s + 0.963·69-s + 1.18·71-s − 1.28·73-s + 0.337·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.592861706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.592861706\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.40717296193690, −13.91236199255518, −13.31131077239865, −12.80509042894707, −12.41274357874342, −11.89792392995244, −11.31474732435570, −10.61723406050728, −10.14127099548226, −9.799045315401731, −9.217995548207968, −8.566357333053405, −8.097122232801872, −7.623562433960880, −7.057706316892495, −6.631080327716618, −5.675963467391551, −5.250223254551096, −4.777873492520047, −3.968648291907307, −3.336148371978817, −2.742457992386439, −2.323299250607315, −1.248345531066765, −0.6881227333801396,
0.6881227333801396, 1.248345531066765, 2.323299250607315, 2.742457992386439, 3.336148371978817, 3.968648291907307, 4.777873492520047, 5.250223254551096, 5.675963467391551, 6.631080327716618, 7.057706316892495, 7.623562433960880, 8.097122232801872, 8.566357333053405, 9.217995548207968, 9.799045315401731, 10.14127099548226, 10.61723406050728, 11.31474732435570, 11.89792392995244, 12.41274357874342, 12.80509042894707, 13.31131077239865, 13.91236199255518, 14.40717296193690
